A complex number $z_1$ has a magnitude $|z_1|=3$ and an angle $\theta_1=20^{\circ}$. Express $z_1$ in rectangular form, as $z_1=a+bi$. Round $a$ and $b$ to the nearest thousandth. $z_1 = $
Solution: The Strategy A complex number of the form $z={a}+{b}i$ has: A magnitude of ${|z|}=\sqrt{{a}^2+{b}^2}$. An angle of ${\theta}=\arctan\left(\dfrac{{b}}{{a}}\right)$. [How did we get these equations?] Therefore, given the absolute value ${|z|}$ and angle ${\theta}$, the parts ${a}$ and ${b}$ are given by the following two equations: ${a}={|z|}\cos{\theta}$ ${b}={|z|}\sin{\theta}$ [How did we get these equations?] Finding $a$ For ${|z_1|}={3}$ and ${\theta_1}={20^{\circ}}$, we can find ${a}$ as follows. $\begin{aligned}{a}&={|z_1|}\cos{\theta_1} \\\\&={3}\cos{20^\circ} \\\\&={2.819}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={|z_1|}\sin{\theta_1} \\\\&={3}\sin{20^\circ} \\\\&={1.026}\end{aligned}$ Summary We found that ${a}={2.819}$ and ${b}={1.026}$. Therefore, $z_1=2.819+1.026i$.